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Ladies and Gentlemen, I´d like to welcome you to our course "Physical Chemistry". My
name is Dr. Lauth and today's topic is: How to express a gas in numbers microscopically?
We have seen that the state of a gas can mathematically be described by the equation of state p * V
= n * R * T We will now try to explain the properties of a gases on the basis of model
concepts. The conceptual model, which we will discuss
today, is called "kinetic theory of gases". The ideal gas law describes a gas macroscopically
-- gives a correlation between its state variables mathematically. The kinetic theory of gases
describes a gas microscopically - a conceptual model.
Previously, a gas was a black box to us. We were able to make measurements on it and could
determine macroscopic state variables such as pressure p, volume V, temperature T, viscosity
eta, lambda thermal conductivity, diffusion constant D etc..
Now we want to set up a model which allows us to get a deeper understanding of a gas
and which relates these macroscopic (measurable) quantities to microscopic quantities (which
measurement is not as easy) These microscopic quantities are essentially
molecular properties of the gas molecules, for example the number N of particles, the
mass m of the particles, size of the particles sigma, velocity v of the particles, etc.
In kinetic theory of gases in the simplest form, it is assumed that a gas consists of
many mass points. The volume of these mass points is very much smaller than the volume
occupied by the gas. These mass points have no interaction with
each other, unless they collide. The particles are in constant motion, collide with each
other and with the container walls. Because there are very many particles, and
because their velocities and energies are constantly changing, we have to deal mean
values. Mean values are represented by a horizontal bar on the symbol or two square brackets.
The mean value of the velocity (v bar) is the sum of the velocities of all particles
divided by the number of the particles. Let me anticipate the gist of the kinetic
theory of gases: The temperature of a gas is directly proportional to the average translational
energy of its particles. In fact, "temperature" is just another term
for the average translational energy of the gas. In formula notation: E bar equals three
halfs Boltzmann's constant times temperature. A particle has three translational degrees
of freedom to move in space. So this results in one half k times T for each direction.
The pressure is calculated by the following formula from microscopic sizes.
1/2 mv ² corresponds to the kinetic energy and is divided by the volume.
So pressure is a measure of energy density. At room temperature (about 300 K), the average
translational energy of any gaseous particle is 6.2∙〖10〗^(-21) J (0.039 eV, 3.7 kJ
/ mol). Comparing two gases, for example oxygen (O2)
and hydrogen (H2) under standard conditions, we generally state different densities.
Oxygen (which has got a density of1.3 g / L -- about the same as air), is substantially
less dense than hydrogen (0.08 g / L). Since both gases are at the same temperature, they
both have the same kinetic energy of translation, namely 3.7 kilojoules per mol
Same temperature - same kinetic energy. Kinetic energy can be expressed as 1/2 m v
². If this has to hold for both gases, there must be a mathematical relation between the
velocities and the masses. We arrive at so-called Graham's law, which
states that the mean velocities are inversely proportional to the square root of the masses.
Because the mass of hydrogen is 16 times smaller than the mass of oxygen,
is the average velocity of hydrogen is four times (root of 16) larger than the average
velocity of oxygen. Oxygen has an average velocity of 444 m / s, hydrogen is faster
than 1700 m / s The kinetic theory of gases is closely linked
with the two scientists Maxwell and Boltzmann. They developed this famous equation which
describes the velocity distribution in a gas. To put it bluntly the formula shows how much
particles move at a certain velocity. On the x-axis we plot the velocity v and on the y-axis
we plot the number of particles... ... and now comes the clarification -- number
of particles that move in a velocity interval dv.
You see that most oxygen particles travel (blue line) at a velocity of about 400 meters
per second. There are also very fast particles, moving
at over 1000 meters per second. These will have a high level of energy and are important
for starting certain kinetic process. Equally there are particles that move very
slowly. Mathematically speaking, the Maxwell-Boltzmann
velocity distribution is a product of a parabolic function (v ²) with an exponential function
- with a maximum at about 400 meters per second for oxygen.
For hydrogen (red curve) the Maxwell-Boltzmann distribution is much flatter, but also shifted
to the right. This velocity distribution is not symmetric but shifted slightly to the
right. (not comparable to a Gaussian bell shaped curve)
There are three velocities that deserve our special attention. Once the most probable
velocity v sub m - this corresponds to the maximum of the Maxwell-Boltzmann distribution.
By curve sketching we can easily find the extremum by setting the first derivative to
zero. We obtain the expression square root of (2 * R * T over M).
The most important velocity is the average velocity v bar. (slightly right of the maximum)
v bar equals root (8 * R * T / M * pi) There is another important velocity - it is
the root mean squared velocity (RMS) This is the velocity of the particles that exhibit
exactly the average energy. In these three formulas you can either use
R / M (molar mass over gas constant) or the ratio k / m (Boltzmann constant over mass
of a particle) . Anyway, it is important to always use consistent units -- I recommend
SI units. The particles collide with the wall and with
themselves. If you want to calculate the number of collisions of a particle with other particles,
then you have to deal with the concept of collision cross section.
We need to mathematically define clearly what is a collision: consider a particle B moving
from left to right with a velocity v. Around the velocity vector of particle B, imagine
a kind of cylinder, this collision cylinder has the cross section sigma (collision cross
section). If another particle is located with its center
of mass in this cylinder, then this particle experiences a collision. There will be no
collision with Particles A prime, as its center of mass is outside the collision.
In contrast, Particle A will collide with B because its center of mass is within the
collision cylinder. We now simply have to count the number of particles in the collision
cylinder to get the number of collisions. The collision cylinder has the basal face
sigma and the length proportional to v bar ie the volume v bar times sigma. We introduce
root 2 as a correction factor for the relative movement of the particles and multiply by
the particle density N over A. This equation gives z - the collision frequency of a particle.
The collision cross section sigma can be estimated geometrically as pi times d ² (d is the particle
diameter). With the values for argon at standard conditions: velocity 400 m / s, particle density
of 2.4 times 10 to 25 particles per cubic meter;
Atomic radius of 0.17 nm, collision cross section of 0.36 nm ², results in a collision
frequency of 4.9 GHz. Each particle collides in one second with 4.9 billion other particles.
This brings us to one of the most important parameters in kinetic theory of gases, the
mean free path. Though a gas particles is very fast (about
400 m / s), it´s always hindered in progression, ad it collides with many other particles.
A gas particle zig-zags its way in a so called "random walk" - through the gas. The mean
distance traveled by a particle between two collisions is called mean free path lambda
bar. v bar is the horizontal distance that is traveled
in one second. Z is the number of collisions in one second. So lambda bar is equal to v
bar over z. Eventually, we obtain this equation.
With the known data of argon at standard conditions, we obtain lambda bar equal 82 nm
A gas particle moves about 400 times as far as its own diameter, before it collides with
another particle (and so changes direction and velocity).
By comparison, the average distance between two gas molecules in argon at standard conditions
is 3.5 nm (drawing is not to scale) The particle density N / V can be calculated
as p / kT by using the ideal gas law, Plugging this into our equation we find that the mean
free path is proportional to temperature and inversely proportional to pressure.
The gas particles also collide with the system´s walls. We will do an estimation for the frequency
of wall collisions z sub w : The higher the gas density, the more collisions will occur,
z sub w will be proportional to the particle density N / V.
Collisions will also increase with the mean velocity of the gas particles, whereas the
collision cross section will be irrelevant, because even the smallest particles will hit
the wall. Eventually we arrive at the following equation,
a correction factor of 1/4 in allowing for not every gas particles flying towards the
wall. Plugging in the numbers for Argon at standard conditions, we calculate that every
square-meter wall is hit by 2.4∙〖10〗^25 particles every second
Pressure is a consequence of these collisions. Consider an elastic collision of a gas particle
with the wall. The x-component of the particle´s momentum is m * vx before the collision and
-- m (vx) afterwards. The momentum transfer 2mvx results in a force
on the wall. Dividing the average force by the wall´s surface we obtain pressure.
The mean free path and the mean velocity of gas particles play a major role in vacuum
technology, for instance mass spectrometry. To separate different ions according to their
mass in a mass filter, these ions must not collide with other particles on the way from
ion source to ion detector. We need a pressure which allows for a mean free path of the ions
which is greater than the dimension of the mass spectrometer device.
(so we need high vacuum or ultra high vacuum). Analyzing air in a Mass spectrometer we get
the following mass spectrum : 28 - the molar mass of nitrogen 32 - the molar mass of oxygen
16 - 18 atomic oxygen - water from the humidity kinetic theory of gases is also helpful to
various methods of surface coating such as "physical vapor deposition" or "chemical vapor
deposition" Furthermore, the mean free path plays an important
role in conductive heat and mass transfer (that is: diffusion) in gases.
(Let´s summarize kinetic gas theory) A gas is made up of particles, moving at high velocity;
the volume of the individual particles being much smaller than the total volume of the
gas. Collisions of the particles to the container wall generate pressure.
Collisions between the gas particles result in a mean free path which -- for example - limits
the rate of mass and heat transport. The macroscopic state variable temperature corresponds to
the average translational energy of the particles. The pressure, we can measure macroscopically,
corresponds to the energy density. The mean velocity of a typical gas at standard condition
is a few 100 meters per second and is inversely proportional to the square root of the mass.
The mean free path depends on the bulkiness of the particles (quantified by collision
cross section sigma) and the particle density. The number of wall collisions is proportional
to the gas density and the average velocity. Thank you for watching.